Introduction

The simulations below were obtained using the non-linear shallow-water solver of Gerris. This numerical model provides solutions of the mathematical equations describing the flow of a "single layer" of fluid. It is based on the assumption that the wavelength of the tsunami waves is large compared to the depth of the sea. As can be seen on the accompanying animations, the wavelength of tsunami waves can vary between several tens of kilometres in deep water down to a few tens of metres in shallow water. The assumption that their wavelength is long compared to the depth of the sea is verified in most areas.

Numerical models describe the solution by sampling it at given intervals. The spacing between these samples (the "spatial resolution") controls how accurate the solution is. Short waves (near the coast) need to be described with very short sampling intervals (high resolution) whereas long waves (in deep water) can be described with longer sampling intervals (lower resolution). Gerris automatically adjusts this sampling interval to guarantee accurate solutions everywhere. For the simulations below the sampling interval varies automatically between about one kilometre in deep water down to 60 or 15 metres near the coastline. This allows to capture accurately both the long-distance tsunami wave propagation (for distant earthquakes such as Kuril for example) as well as very detailed local inundation on the coasts of Wallis and Futuna. This "adaptive" numerical method has been validated extensively both using theoretical and experimental test cases [Popinet, 2011] as well as real-world tsunamis [Popinet, 2012].

Aside from the choice of spatial resolution, the accuracy and reliability of the modelled scenarios depend also on the quality of the input data. The bathymetry and topography data are important as they control both the deep-water propagation as well as details of local inundation. For Futuna, we had only limited data for the local topography (contour lines at 0, 2.5, 5, 10, and 20 metres) and high-resolution multibeam data but only for depths below 10 metres. We had no data on the fringing reef. The reef is particularly important as the shallow water in this area can break and dissipate tsunami waves before they reach the coastline. To fill this gap, we digitised a single line for the contour of the reef from navigation charts and set its depth to -1 metre everywhere. This is clearly a very crude approach but was the best we could do.

For Wallis, we had acquired high-resolution GPS data for most land areas close to the coastline (on the main island only) and also had access to high-resolution multibeam data for some small areas in the lagoon, as well as for depths below 10 metres for the outer reef wall. The influence of the reef and lagoon is of course even more important for Wallis than for Futuna. For the outer reef platform, we used the same approach as for Futuna, using digitised reef outlines from navigation charts and setting a constant depth of 0 metres. We also added some contours in the lagoon obtained by analysis of ocean colour on satellite images. While the outer reef platform can be considered adequately represented by this approach, the complex bathymetry of the lagoon is very crudely represented and this needs to be taken into account when interpreting high-resolution inundation results. While we had more topographic data for Wallis than for Futuna, we consider the terrain model for Wallis to be significantly less reliable than that of Futuna, because the lagoon bathymetry is much more complex.

The other important input to the model is the initial wave field generated by earthquakes. All the sources below are based on the "Okada elastic deformation model" which assumes simple elastic caracteristics for the deformations of the sea floor due to fault ruptures. The corresponding vertical displacement of the seafloor is assumed to translate directly into a vertical motion of the sea surface. This initial vertical deformation is used as input for the tsunami model. This is a very standard approach which has been shown to be able to reproduce historical tsunamis, with compatible estimates of the intensity of the triggering earthquakes. From a tsunami generation perspective, the most important parameters are the total energy released (i.e. the magnitude), the fault orientation and the "dip angle" of the fault plane. The dip angle is important because it controls the amount of energy released as a vertical displacement of the sea surface. For the scenarios below, these parameters are estimated based on our current knowledge of the particular faults concerned.